Sahil Singla Carnegie Mellon University Joint work with Euiwoong Lee 26 th June 2017 TwoStage matching problem Graph Edges Appears in Two Batches Stages Appears in Stage 1 ID: 611388
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Slide1
Maximum Matching in the Online Batch-Arrival Model
Sahil Singla (Carnegie Mellon University)Joint work with Euiwoong Lee
26th June, 2017Slide2
Two-Stage matching problem
Graph Edges Appears in Two Batches/ Stages
Appears in Stage 1
Pick Matching
in (Unknown )Unselected Edges Disappear Appears in Stage 2Select in s.t. is a MatchingGoalMaximize size of Competitive Ratio:Greedy is Half Competitive
2
Can we beat half?Slide3
The Z Graph
Graph Appears in Two Batches
Appears
Pick Matching
in
(Unknown )Unselected Edges Disappear AppearsSelect in s.t. is a MatchingGoalMaximize size of
3
Do we Pick Edge in
?
Pick
w.p
.
Case 1
: E[
Alg
]=
& OPT=1
Case 2
:
E[
Alg
]=
&
OPT=2
Fractional Matching?
Easier than Integral
or
Case 1
Case 2Slide4
Our results
Theorem 1
:
For
Two-Stage Integral Bipartite
Matching, There Exists a Competitive Tight Algorithm. 4
Theorem 2: For
Two-Stage Fractional Bipartite Matching, There Exists an Instance Optimal
Competitive Algorithm.Instance Optimal: Given returns s.t. Gets for every For every Alg,
where ALG
Slide5
Prior Work
5Online ArrivalSingle arrival in each step (linear # stages)Immediate & Irrevocable decisionsVertex Arrival or Edge ArrivalSemi-Streaming Arrival
decisions postponed
Vertex Arrival or Edge Arrival
Two-Stage Stochastic Optimization
Costs change every stageArrival from a known distribution Slide6
OUTLINE
Multi-Stage MatchingExamples & Special CasesProof Idea: Fractional Bipartite Matching
Proof Idea: Integral
Bipartite
Matching
Extensions and Open Problems6Slide7
Randomly Pick Max Matching?
7Find a Max Matching in
Pick it Randomly, and
Nothing
Otherwise
What if Multiple Max Matchings? Which one to pick?With how much probability? Graphs Known Where For Every Max Matching , Randomly Picking gives
Slide8
has A Perfect Matching
Suppose
has a Perfect Matching M
Every vertex with an incident
edge in is matched in MPick M w.p. , and Nothing OtherwiseOptimally Augment in Stage 2How to Prove ? 8
Lemma
:
Above algorithm is Competitive for Two-Stage Integral Bipartite Matching. Slide9
Primal-Dual Framework
9Offline Bipartite Matching LP
max
min
s.t.
s.t.
Opt Solution Certificate For
Show
feasible
s.t.
-Approx Solution Certificate For
Show
-feasible
s.t
.
i.e.,
Slide10
has A Perfect Matching
ALGORITHMPick M w.p
.
, & Optimally Augment in Stage 2Set = 1 when is picked in Stage 1Set = 1 when is picked in Stage 2 Set
10
Lemma
: Above algorithm is
Competitive.
Certificate:
s.t.
&
Set
when
matched
in Stage 1
Set
to be
optimal vertex
cover
for Stage 2, where
Set
Slide11
has A Perfect Matching
Analysis
: Since
-Feasibility
: Case analysis ,
Both
in
:
Both not in
:
Only
in
:
11
Q.E.D.Slide12
OUTLINE
Multi-Stage MatchingExamples & Special CasesProof Idea: Fractional Bipartite Matching
Proof Idea: Integral
Bipartite
Matching
Extensions and Open Problems12Slide13
Two-Stage Fractional Matching
13
Theorem 2
:
For
Two-Stage Fractional Bipartite Matching, There Exists an Instance Optimal Competitive Algorithm.Proof Idea:
Construct an LP on
that maximizes
Gets for every For every ALG, where ALG
Here
OPT(
)
Slide14
A New LP
14
max
s.t
.
Instance Optimality:
Gets
for every
For
every
ALG
,
where
ALG
Ques:
Is
?
Let
Slide15
OUTLINE
Multi-Stage MatchingExamples & Special CasesProof Idea: Fractional Bipartite Matching
Proof Idea: Integral
Bipartite
Matching
Extensions and Open Problems15Slide16
is Expanding
Suppose
is
ExpandingEvery has neighborsCan pick a random matching s.t. & we have &
16
Here
Algorithm
Pick
M
w.p
.
, & Optimally Augment in
Stage 2
Analysis
Set
&
for
when
picked
For any
case-by-case
show for every edge
Slide17
Two-Stage Integral Matching
17
Theorem 1
:
For
Two-Stage Integral Bipartite Matching, There Exists a Competitive Tight Algorithm.
Algorithm
:
Construct a Matching Skeleton of Partition into several Expanding Bipartite SubgraphsRandomly Pick a Max Matching in each Bipartite SubgraphOptimally Augment in Stage 2Proof: Show s.t.
where
for every edge
Slide18
Bipartite Matching Skeleton
18Algorithm :Construct a Matching Skeleton of
Randomly pick a Max Matching in each bipartite subgraph
Optimally
augment
in Stage 2 Goel-Kapralov-KhannaDecompose into is expanding, where
No edge
to
No edge to for Algorithm
Select
uniformly
pick
if
Analysis
Set
&
for
For
any
show for every edge
Slide19
OUTLINE
Multi-Stage MatchingExamples & Special CasesProof Idea: Fractional Bipartite Matching
Proof Idea: Integral Bipartite
Matching
Extensions and Open Problems
19Slide20
Extensions
Theorem 3
:
For
Two-Stage Fractional General
Matching, There Exists a Competitive Algorithm. 20
Theorem 4
: For s-Stage Integral General Matching
, There Exists a Competitive Algorithm. Slide21
General Matching Skeleton
21Edmonds-Gallai Decomposition
Proof Idea:
Run
Bipartite
Algo for Pick Matching in synchronously with Distribute
duals to
vertices & odd-components
Show for any : for every
has
vertex from each odd component
Slide22
Open Problems
22
Problem 1:
For
s-Stage Integral Bipartite Matching, Does There Exist an Algorithm That Beats Half by a Constant?Problem 2: For Two-Stage Integral General Matching, What is the Tight Competitive Ratio?
We showed it’s
and
Slide23
Open Problems
23
Problem 3:
Any Natural Online Problem With
Competitive Algorithm in s-Stage Online-Batch Arrival Model?
Not True For
Online Set Cover
Online Facility LocationOnline Steiner TreeUnrelated Load Balancing (makespan minimization)Slide24
summary
Fractional Bipartite MatchingInstance optimal for two stagesIntegral Bipartite Matching competitive for two-stages
Integral General Matching
competitive
for
s-stage sOpen ProblemsBeat half for linear # stages?Other interesting multistage problems? 24Questions?Slide25
references
L. Epstein, A. Levin, D. Segev, and O. Weimann. Improved bounds for online preemptive matching. STACS’13A. Goel, M. Kapralov, and S. Khanna. `On the communication and streaming complexity of maximum bipartite
matching’. SODA’12D. Golovin, V. Goyal, V.
Polishchuk
, R. Ravi, and M.
Sysikaski. `Improved approximations for two-stage min-cut and shortest path problems under uncertainty’. Math Prog’15R. M. Karp, U. V. Vazirani, and V. V. Vazirani. `An optimal algorithm for on-line bipartite matching’. STOC’90L. Lovasz and M. D. Plummer. `Matching Theory’. Ann Disc Math’86A. Mehta. `Online matching and ad allocation’. TCS’12.C. Swamy and D. B. Shmoys. `Approximation algorithms for 2-stage stochastic optimization problems’. SIGACT’0625